Reachability and Büchi Games - Rich Model Toolkit
Reachability and Büchi Games - Rich Model Toolkit
Reachability and Büchi Games - Rich Model Toolkit
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0-Attractor<br />
To show W 0 = Attr 0 (F) <strong>and</strong> W 1 = S \Attr 0 (F), we construct<br />
winning strategies for Player 0 <strong>and</strong> 1.<br />
Proof.<br />
Attr 0 (F) ⊆ W 0<br />
We prove for every i <strong>and</strong> for every state s ∈ Attr i 0(F) that Player 0<br />
has a positional winning strategy to reach F in ≤ i steps.<br />
◮ (Base) s ∈ Attr 0 0(F) = F<br />
◮ (Induction) s ∈ Attr i+1<br />
0 (F)<br />
If s ∈ Attr i 0(F), then we apply induction hypothesis.<br />
Otherwise s ∈ ForceNext 0 (Attr i 0 (F))\Attri 0 (F) <strong>and</strong> Player 0 can<br />
force a visit to Attr i 0 (F) in one step <strong>and</strong> from there she needs at<br />
move i steps by induction hypothesis. So, F is reached after a<br />
finite number of moves.
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